Degree, multiplicity, and inversion formulas for rational surfaces using u-resultants

Proper Reparametrization of Rational Ruled Surface

“…If a surface is not proper, then for a generic point on the surface, there exist a fixed number of parametric values corresponding to this point [3,5,7]. This fixed number is called the improper index of the parametrization (1), denoted by IX(P).…”

“…This fixed number is called the improper index of the parametrization (1), denoted by IX(P). The improper index of (1) can be found by computing the u-resultant [3] or by computing the Gröbner basis of (2) [5].…”

“…In the case of algebraic surfaces, we can determine whether a surface is proper using the u-resultant [3] or the Gröbner basis [5]. However, the problem of finding a proper reparametrization for an improper rational parametrization of an algebraic surface is open in the 1) This paper is partially supported by a national key basic research project of China.…”

An Implicitization Algorithm for Rational Surfaces with no Base Points

“…The computation of the resultant is not a trivial task (Gelfand et al, 1994;Cox et al, 1998) and very often leads to expressions spoiled by extraneous factors (Manocha and Canny, 1992). For instance, the Macaulay method (Macauley, 1923) proposed by Chionh and Goldman (1992) requires a polynomial division for eliminating the extraneous factor. To obtain the implicit equation, the general methods of this class have so far introduced an intermediate expression of a higher degree than the expected final result.…”

Inherently improper surface parametric supports

“…These facts are surveyed in [6]. Though correcting an improper surface rational parametrization is difficult or impossible, there are practical algorithms to detect one [2,5].…”